From Gold Coins to Cadmium Light

On this page, we will take a brief look at some of the complications encountered
when attempting to define units of measurement with high accuracy.

Time and Length

The second is defined as the time taken by 9,192,631,770
oscillations of the microwave radio frequency produced by an atom
of Cesium-133 when the electrons in that atom are in the ground state, except
for one that has emitted this radiation by making the transition from the
upper hyperfine level of this state to the lower one. This definition was
chosen to make the length of the SI second the same as that of a second of
Ephemeris Time and so when this
second began to be used in civil timekeeping (the changeover to "atomic time"),
the use of "leap-seconds" became necessary immediately.

The 1952 definition of the second of Ephemeris time was based on the instantaneous
length of the tropical year being 31,556,925.9747 seconds at the beginning of the year 1900,
which was considered to be Noon GMT of December 31st, 1899, but according to Ephemeris Time
rather than civil time. However, while the length of the year at that time was the basis for
the standard, the length of the second was not 1/86,400 of the length of the day at that time,
but was rather based on the average of the length of the day from 1750 to 1892 as previously
worked out by Simon Newcomb.

At one time, the metre was defined as 1,650,763.73 wavelengths
of the orange red line in the spectrum of Krypton-86 which corresponded to
the radiation emitted by an electron moving from the orbital

2 p to the orbital 5 d
10 5

in an unperturbed fashion.

However, a scientist proceeded to measure the
speed of light by performing an accurate measurement of the ratio between
the wavelengths (and/or frequencies) of these two types of radiation. In
Zen-like fashion, this convinced those responsible for the standards of
the fundamental absurdity of the situation, and so now the definition of the
second stands, but the definition of the metre has been replaced; now,
the metre derives from the second, through the speed of light, which is, by
definition,

8
2.99792458 * 10

metres per second.

This leads to the length of a metre being approximately 30.6633189885
wavelengths of the Cesium-133 microwave radiation noted above: the fact that the
wavelength is not a tiny fraction of a metre is why it had previously been
considered more suitable to a standard of time, for which a radio frequency which
can be manipulated by electronics is more accessible than an optical frequency, than
one of distance.

The speed of light in a vacuum can also be given in units of the
uniform US/British inch of 2.54 centimetres, which leads to light travelling
186,282 miles, 698 yards, 2 feet, and 5 21/127 inches every second in a
vacuum.

Note that it is more convenient
to measure the length of waves of light through interference fringes,
and the time between oscillations of
radio waves through electrical circuitry, however. So the dual
definitions allowed both time and distance to be more accurately defined.

Incidentally, as far back as 1927, a definition of the metre in terms of
a light wavelength existed, but that definition was based on a line
in the spectrum of cadmium: the length of
the metre was defined as 1,553,164.13 times the wavelength of the
6436.4696 Ångström cadmium red line.

A. A. Michaelson had found the cadmium red line to be particularly
monochromatic; he had made a measurement of the
International Prototype Metre in 1892 which indicated the length of
the metre as 1,553,163.5 times the wavelength of the
cadmium red line, the wavelength of which was taken to be 6436.472
Ångström units.

The figure used in the 1927 standard of 1,553,164.13 wavelengths per
metre corresponds closely to a wavelength of 6436.4696 Ångströms for
the cadmium red line; that the cadmium red line is exactly 6436.4696 Ångström
units long was adopted as a standard for the definition of the international Ångström
unit in 1907, based on measurements in 1906 by Benoit, Fabry, and Perot of the length
of the metre in terms of the cadmium red line.

Cadmium, however, has eight stable isotopes, the most common of which,
Cadmium-114, has a natural abundance of 28.73%. so the accuracy of a standard
based on natural cadmium would be limited by the small variations
in the wavelengths of the same spectral line between different isotopes.

Prior to the use of Krypton-86 for the standard length of the metre, another
possibility that was considered was to use Mercury-198. This isotope of mercury
was created in 1946 by bombarding gold with neutrons, as gold, like aluminum, has
only one stable isotope, and this method of creating isotopically-pure mercury was
simpler than attempting to separate the isotopes of mercury by their miniscule
difference in weight. Its 5460.753 Ångström spectral line was the one considered
for use as a standard. Also, a current secondary standard for the metre
is an iodine-stabilized helium-neon laser, the light from which has a wavelength of
6329.9139822 Ångströms.

It is necessary at this point
to add that 3515.3502 wavelengths of the cadmium red line would, by the
1927 definition, subtend some 2.2633475317 millimetres (as opposed to 2.2633485174
millimetres), which seems to indicate a discrepancy in the definition of the
Potrzebie. As a little arithmetic shows that this definition of the Potrzebie
would lead to a metre of 1,553,163.45 wavelengths of the
red line of cadmium in length, it seems apparent that Donald Knuth, then a
19-year old high school student, had converted from millimetres to wavelengths
of cadmium light using a reference giving the 1892 figure.

In my efforts to sort out the mystery of that discrepancy, which led me to
finding out about the 1892 figure, I had encountered a biographical essay on
A. Michaelson by Robert A. Millikan giving 6438.472 Ångströms as the
wavelength of the red line of Cadmium as determined by Michaelson which would indeed
lead to a standard of about 1,553,163.5 wavelengths per metre. In the essay, the resulting
standard as 1,555,165.5, which I took to be a misprint. Thus, the mystery of the difference
between the length of the Potrzebie as defined in terms of the metre and that as defined in
terms of the cadmium red line appeared to be solved, and further searching led me both
to confirmation that 1,553,163.5 was the figure arrived at by Michaelson in 1892 and
to find that the later standard derived from the 1906 measurements of Benoit, Fabry,
and Perot as noted above.

It may also be noted that in 1964, an agreement was reached between
the U.S. and Britain to define the inch as 2.54 centimetres.

Prior to 1964, the inch was defined in the U.S. on the basis that a
metre was exactly 39.37 inches long, which led to the inch being
about 2.540005 centimetres long, and in Britain the inch
was 2.539997 centimetres in length. (One older reference gives the metre
being 39.37079 inches, and the inch therefore being 2.539954 centimetres
in length.)

Or is the British inch 2.539996 centimetres? There are sources giving both values
for the British inch. I found the 2.539997 figure in the Encyclopedia Britannica article
on weights and measures, while the 2.539996 figure occurs in news stories from 1938 and
1958 about the ongoing negotiations to establish agreement between Britain and the United
States on the inch of 2.54 centimetres, so I would have been inclined to believe the
2.539997 figure to be the more accurate one. Yet another source gives the length of the
inch based on the British "Bronze Yard #11" to be 2.53999944 centimetres.

Unlike the case of the U.S. inch, where the inch was defined so that 39.37 inches equalled
one metre, so that the U.S. inch must therefore be 2.54000508001016002032004064.. centimetres,
the British inch was defined by a separate physical standard, so there is no inherent exact
ratio between the British inch and the metre.

A 1928 scientific paper, titled "A New Determination of the Imperial Standard Yard to the
International Prototype Metre", by Sears, Johnson, and Jolly, gave the ratio of 1 metre to
39.370147 British inches, which gives rise to an inch of 2.5399955596... centimetres. Incidentally,
the International Prototype Metre was made in 1872, while the Imperial Standard Yard dates back to
1855.

An earlier measurement, from 1895, gave the length of the metre as 39.370113 British
inches, leading to a British inch of 2.53999778969... centimetres. That rounds to 2.539998,
and so it cannot alone be the source of the other value; despite the fact that one source
notes that the 39.370113 figure was the de facto standard for the length of the
British inch for many years.

The Imperial Standard Yard is made from a bronze alloy, 82% copper, 13% tin, and 5% zinc,
and it is defined to have its standard length at a temperature of 62 degrees Fahrenheit;
the International Prototype Metre is made of 90% platinum and 10% iridium, and its length
is valid at 0 degrees Celsius (32 degrees Fahrenheit, or the freezing point of water).

Incidentally, the Imperial Standard Yard was noted as having shrunk by one part per million in a span
of 20 years, calling its accuracy into question.

It may be noted that the Pyramid Inch was claimed to be 1.00106 English inches,
so that would make it about 2.5426894 centimetres long. The Pyramid inch was said to
be 1/25th of a royal cubit.

In fact, an ordinary cubit, about 18 inches long (so they
were at least right that cubits related better to Imperial measure than to the metric
system) was divided into six spans (each three inches long), which were in turn divided
into four digits (each 3/4 of an inch; and, indeed, the keys on our typewriter keyboards
have 3/4 of an inch spacing even today). A royal cubit is seven spans instead of six,
and so, nominally, it should be 21 inches long, but then standards of measure were less
accurate in those days. The royal cubit was used in the construction of the Great Pyramid;
thus, its sides had a rise of one royal cubit for a run of five and one-half spans;
which, multiplied by four, gives 3 1/7, giving the appearance that pi is involved in the
construction of the Pyramids.

In fact, though, serious archaeologists and historians now know that the Egyptian
royal cubit was about 52.6 centimetres (give or take 3mm), or 20.7 inches - so it was
indeed near to 21 inches, and actually somewhat smaller, and thus not 25.0265 inches
long. To the extent, therefore, that such a thing as an ancient Egyptian inch has any
meaning, therefore, it would be about 98 4/7 percent of an inch, not 1.00106 inches,
in length.

While the Egyptian measure relates to Imperial measure by a factor of about .986,
the currently accepted value for the Roman foot makes it .971 feet long; thus, while
the foot grew on its way to Britain, in the middle, as it passed through Rome, it
shrank. And the Romans did divide the foot into 16 digits as well as 12 inches, and so
linking the cubit to the inch as I have done is legitimate.

I think it is unfortunate that they missed their chance to define the
inch as being about 2.540002 centimetres in length, so that the
diagonal of a square 152 inches on a side would be exactly 546 centimetres,
or the diagonal of a square 273 centimetres on a side would be exactly
152 inches. After all, supporters of the metric system have always
criticized the Imperial system as irrational; and it would be convenient
if having two systems of measurement allowed one, by using both of them, to
measure exactly both the sides of a square and its diagonal.

Of course, lengths of 152 inches and 273 centimetres are somewhat unwieldy.
However, as rulers measuring inches are often divided in tenths of an inch,
one could relate 15.2 inches to 273 millimetres. But inch rulers are more
often divided into sixteenths of an inch.

If one were to use the same method to relate the sixteenth of an inch
to a millimetre, defining the inch as about 2.5399946 centimetres would lead
to a square 284 millimetres on a side having a diagonal of 15 and 13/16 of
an inch.

However, failing changes in our systems of measurement, one can always
simply make use of the fact that 20 squared is 400, 21 squared is 441, and
29 squared is 841 to come reasonably close to a 45 degree angle and still
use exact distances.

Also, even if redefining the inch is excluded, if only a single unit of
measurement is used, comparable ratios to approximate the square root of two
would be 239:169 and 577:408, which are in error by 0.000875 percent and 0.000150
percent respectively, while, using the inch of 2.54 centimetres, the ratio
152 inches to 273 centimetres approximates the square root of two with an error
of 0.000078 percent.

The diagonal of a square 273 centimetres on a side is 386.0803025278548... centimetres,
while 152 inches of 2.54 centimetres each are 386.08 centimetres, so there is an
excess of 0.0003025278548... centimetres. The older U.S. inch, such that 39.37
inches equal one metre, is still in use for survey purposes, and this inch is
equal to 2.54000508001016... centimetres. If, of the 152 inches of the diagonal,
59 and 9/16 of those inches were measured using the older U.S. inch, and the other
92 and 7/16 of those inches were measured using the current inch of 2.54 cm, an
even closer approximation to the square root of two would be obtained.

A more approximate measurement of the diagonal of the square can be obtained
using much simpler numbers. The diagonal of a square 9 centimetres on a side
is 5.0109929... inches in length. For comparison, the diagonal of a square 7
centimetres on a side is 9.8994949... centimetres in length. The discrepancy,
in addition to being in the opposite direction, is very nearly 3.6 times as
large. So, the diagonal of a square 39.4 centimetres on a side, which is
55.7200143574999... centimetres, is very close to 10 centimetres (about the
diagonal of 7 centimetres) plus 18 inches (about the diagonal of the other 32.4
centimetres) since 18 inches is 45.72 centimetres.

Since these are all even numbers, we can note that 9 inches plus 5 centimetres
is approximately the diagonal of 19.7 centimetres.

Since those words were written, it occured to me that there might be another
irrational number that should instead be used to define the relationship between
the centimetre and a modified inch.

If one decides that a computer keyboard should be tilted at an angle of ten degrees,
then while the spacing of circuit traces for the different keys would be, horizontally,
three-quarters of an inch, vertically they would be that distance divided by the cosine
of ten degrees. So, if that could be made something reasonably easy to specify...

With an inch of 2.54 centimetres, the distance in question is 1.93438769564234419687...
centimetres approximately. If, for example, we want that distance to instead be 1.9344 cm
exactly, we would have to change the inch to 2.540016156569... centimetres.

On one page on this site, I note that the Egyptians had a royal cubit of
seven spans as opposed to the regular cubit of six spans. But I note at least
one page that claims the original form of the royal cubit in Egypt was a measuring
unit for measuring diagonals.

Mass (and Weight)

The U.S. pound was redefined as 453.59237 grams in 1959, having previously
been defined on the basis of 2.2062234 pounds equalling one kilogram
exactly, leading to a pound of about 453.5924277 grams, nearly identical
to the British pound.

A pound is 7000 grains in weight, that is, the normal, 453.59 gram pound
used to weigh food. Thus, if you have peas to weigh, you use
this pound, which is called the avoirdupois pound. The troy ounce,
which is used to weigh gold, however, is 480 grains in weight, and
there are twelve troy ounces in a troy pound.

The peak high in 1980 (New York), the peak low in 2001 (London), and the
new record-setting price (New York) are noted.

People writing mediaeval role-playing games should note that due to the
difference in density between gold and silver, if one uses the
historic ratio that gold is 16 times more valuable than silver by weight,
then it is also (approximately) 25 times more valuable than silver
by volume. This will make it possible to determine more accurately
how much treasure your characters can carry in their backpacks, while still
using handy round numbers.

Prior to the devaluation of the U.S. dollar in 1933, which also led
to the abolishment of gold coinage, the U.S. dollar was defined by defining the
Gold Eagle, a coin with a denomination of $10, as being composed of
258 grains of 9/10 fine gold. In U.S. coins, the remaining tenth
was made up of half silver and half copper, which were not
without value themselves.

The Constitution originally defined the dollar as 371.25 grains of silver, and
that is 90% of 412.5. In 1872, the weight of the silver dollar was reduced to
384 grains of 9/10 fine silver, because the price of silver had gone up again relative to that
of gold, and gold was generally accepted as the international standard of monetary value.
However, unlike the previous silver dollar, while this coin was still intended to be
worth nearly a dollar, it was not expected to be worth exactly a full dollar; it was not
token coinage, but it was subsidiary coinage.

If the situation between 1834 and 1872 is used as the basis, then, an attempt can be made
to determine the exact nominal value of pure gold in U.S. dollars.

So, 23.22 grains of pure gold and the remaining 1.15333... grains of copper equal
99.6525252... cents; thus, if we neglect the value of the copper, that works out to one troy
ounce of gold (480 grains) being worth almost exactly $20.60 instead of $20.67.

If silver were worth 1/16th as much as gold, this would make 371.25 grains of silver
worth about 99.58 cents instead of a dollar.

In 1805, the price of copper was considered high at 138 pounds a ton, or about $690 a
ton. A ton being 2000 avoirdupois pounds of 7000 grains, this would be about 2.366 cents
per troy ounce, compared to about $1.29 per troy ounce for silver and $20.60 per troy
ounce for gold. Thus, the ratio of value by weight, historically, had been in the neighborhood
of 50 to 1 between silver and copper, or perhaps higher.

Given the composition of the Gold Eagle above, a half-eagle, with a $5 denomination, would
be composed of 116.1 grains of pure gold, 6.45 grains of pure silver, and 6.45 grains of
pure copper. A Gold Sovereign from Britain, on the other hand, is composed of 113.0015 grains of
pure gold, and 10.2729 grains of copper, for a total weight of 123.2744 grains, or 7.9881 grams.
Another source gives the weight of a British Gold Sovereign as 5 pennyweights, 3 171/623 grains;
one pennyweight is 24 grains, so this is 123 171/623 grains, or approximately 123.2744783306581 grains.
This works out to about 113.0016 grains of pure gold at 22 carats.

If the halfpenny were exactly equal to the cent, making the British pound equal to $4.80 in value, 113 grains of
gold to the pound would imply about 117.7 grains of gold to five dollars; thus, while a pound was not worth quite
as much as five U.S. dollars, it was actually worth about $4.85 at the time of the gold standard. (Actually, the
value of the mint par of exchange used at the time was $4.8665.) While the Canadian dollar was made to be equal
to the U.S. dollar, the Newfoundland dollar was about 1 1/3 cents more than a U.S. dollar, as one would expect
from a dollar equal to 100 ha'pennies.

In that era, other countries also had currencies which were fixed in value in terms of gold, and thus to one
another. Thus, 19.2952 cents, at 4.48036 grains of gold, was the value of a French Franc, a Swiss Franc,
a Belgian Franc, an Italian Lira, a Bulgarian Lev, or a Greek Drachma in those days.

Here is a rough table of coin diametres and values, where the values
are in terms of the value of U. S. currency in precious metals between
1892 and 1933, and the upper value is for silver coins, the lower for
gold coins:

Moving one column to the right increases the weight and value
of the coin by a factor of 1.25, moving one row down doubles the
weight and value of the coin. The sizes have been formed on the basis
of the currencies of several different countries, and thus they tend
to be smaller than the sizes of U.S. coins, and/or larger than the sizes of
British coins of corresponding value (1 shilling = 25 cents,
1 florin = 50 cents, 1 crown = $1.25, 1 pound = $5.00) up to the
year 1919.

Because copper or bronze coins tend to be token coinage,
their size is quite variable; the value ratio of 20:1 for gold and silver
at the same diametre is consistent, but that between silver and copper
can vary from 25:1 to 50:1.

The density of gold is 19.32 grams per cubic centimetre, that of silver
10.5 grams/cc, that of copper 8.96 grams/cc. A ratio of 64:1 in
value by weight would lead to a ratio of 75:1 in value by volume for
silver and copper as metals. Copper coins tend, however, to be a
token currency, that is, a metal form of paper money as it were, and
so the ratio in weight of copper coin to silver coin would be more
like 16:1 or 32:1.

The atomic weight of gold is 196.9665, and its one stable
isotope is Gold-197. An atomic mass unit is

-24
1.6605655 * 10

grams.

Thus, one grain of gold is .064798918... grams of gold, and would contain some

20
1.9811592 * 10

atoms of gold.

Using a 16:1 ratio of value between gold and silver, and a
64:1 ratio of value between silver and copper for a 1024:1 ratio of value
between gold and copper, one finds that U.S. gold coins have a
metal content whose value is that of

25.8 * (.9 + .05/16 + .05/1024) =
24.038847656...

grains of gold per dollar,
one could, if one wished,
define a dollar as the pecuniary value of

21
4.7624784 * 10

atoms of Gold-197, or, as they say,

197
Au
79

contained in a
good delivery gold bar, which is a bar of gold that is at least 99.5% fine,
and which has a mass of approximately 400 troy ounces, ranging from 350 troy
ounces to 430 troy ounces (the actual fineness and the weight being marked on the
bar). 400 troy ounces is about 12,441.4 grams, and the range would be from a low of
10,886.22 grams to a high of 13,374.49 grams.

(Whether or not the avoirdupois pound is a unit of force instead of a unit of
mass, the troy ounce is definitely a unit of mass.)

Liquid Measure

Speaking of the kilogram, I have read news items noting that the international standard
kilogram has experienced changes in its mass, and that an effort was underway to make a more
accurate standard from a sphere of silicon.

The metre and the second can be defined in terms of spectral lines or microwave emissions from
specific substances. Could the gram be physically defined?

Originally, the litre was defined as the volume of a kilogram of water. As well, a litre was
nominally 1,000 cubic centimetres of water. However, because water has a pronounced tendency to
dissolve, at least to a limited extent, almost any other substance, it is not a good material to
use for a physical standard. Thus, for a time, a litre, instead of being exactly 1,000 cubic centimetres
or one cubic decimetre, was instead officially defined as 1.000028 cubic decimetres, because this was, as
near as could be determined, the volume that a kilogram of water actually occupied; this anomaly was
corrected in 1964, and thus now one millilitre (ml) is exactly the same as a cubic centimetre
(often noted by the non-SI abbreviation cc instead of cm3).

Thus, if it were desired to give the kilogram a reproducible physical definition based on water,
it would have to be redefined so that the new kilogram was 0.999972 existing kilograms, and this would
not be acceptable, being too large a change, and affecting the derived units of energy and force, and
hence the various electrical units.

Liquid measure has always been one of the more confusing areas of measurement.

In the United States, a gallon is 231 cubic inches by definition.

In the British imperial system, a gallon was originally defined as the volume of ten pounds of water.
Subsequently, the British gallon was defined in terms of metric units as 4.54609 litres; this was in
1985, after the litre had become exactly a cubic decilitre.

The Imperial gallon is larger than the U.S. gallon, but the Imperial fluid ounce is smaller
than the than the U.S. fluid ounce, due to one difference between the two systems:

the space in the table dividing the part in which the U.S. units are smaller from the part
in which the U.S. units are larger. Often, the tablespoon is approximated by 15 ml in recipies now,
although I've read that in Australia, 20 ml is instead used as the approximation. Thus, one
could speak of a metric scruple of 1.25 ml.